The intersections of the lower central series and the subgroups of finite p-index of Generalized Baumslag-Solitar tree groups

TL;DR

The paper analyzes residual properties of Generalized Baumslag-Solitar tree groups by computing the intersections $\gamma_{\omega}(G)$ of the lower central series and $(N_{p})_{\omega}(G)$ of finite $p$-index subgroups. The authors prove that, for a GBS-tree group $G$, both intersections are generated in a natural, path/vertex-based way: $\gamma_{\omega}(G)$ is generated by the $\gamma_{\omega}$-subgroups of two-vertex subgroups $H_{ij}$, and $(N_{p})_{\omega}(G)$ by the corresponding path-subgroup contributions; the tree case reduces to aggregating segment (path) computations. They provide explicit segment calculations, extend these to trees, and illustrate the method with a detailed example where $\gamma_{\omega}(G)$ and $(N_{3})_{\omega}(G)$ are computed as normal closures of concrete commutator sets. The results give structural insight into residual nilpotence and finite $p$-residual properties for GBS-tree groups, with potential applications to understanding residually finite and torsion-related behavior in these groups.

Abstract

For a Generalized Baumslag-Solitar group $G$ with underling graph a tree, we calculate the intersection $γ_ω(G)$ of the lower central series and the intersection $(N_{p})_ω(G)$ of the subgroups of finite index some power of a prime $p$.

Theorems & Definitions (43)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 2.5: see Varsos
• Proposition 2.6